2cos(x)=cos^2(x)-1

{ "query": { "display": "$$2\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)-1$$", "symbolab_question": "EQUATION#2\\cos(x)=\\cos^{2}(x)-1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.99787…+2πn,x=-1.99787…+2πn", "degrees": "x=114.46980…^{\\circ }+360^{\\circ }n,x=-114.46980…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)-1{\\quad:\\quad}x=1.99787…+2πn,\\:x=-1.99787…+2πn$$", "input": "2\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)-1", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "2\\cos\\left(x\\right)=\\cos^{2}\\left(x\\right)-1", "result": "\\cos\\left(x\\right)=1+\\sqrt{2},\\:\\cos\\left(x\\right)=1-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "2u=u^{2}-1" }, { "type": "interim", "title": "$$2u=u^{2}-1{\\quad:\\quad}u=1+\\sqrt{2},\\:u=1-\\sqrt{2}$$", "input": "2u=u^{2}-1", "steps": [ { "type": "step", "primary": "Switch sides", "result": "u^{2}-1=2u" }, { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "u^{2}-1=2u", "result": "u^{2}-1-2u=0", "steps": [ { "type": "step", "primary": "Subtract $$2u$$ from both sides", "result": "u^{2}-1-2u=2u-2u" }, { "type": "step", "primary": "Simplify", "result": "u^{2}-1-2u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "u^{2}-2u-1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "u^{2}-2u-1=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-1\\right)}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=-2,\\:c=-1$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-1\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-1\\right)}=2\\sqrt{2}$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:1\\cdot\\:\\left(-1\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:2\\sqrt{2}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:1\\cdot\\:1}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}+4\\cdot\\:1\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=\\sqrt{2^{2}+4}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+4}" }, { "type": "step", "primary": "Add the numbers: $$4+4=8$$", "result": "=\\sqrt{8}" }, { "type": "interim", "title": "Prime factorization of $$8:{\\quad}2^{3}$$", "input": "8", "result": "=\\sqrt{2^{3}}", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "primary": "$$2$$ is a prime number, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:2" }, { "type": "step", "result": "=2^{3}" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1RcgsS082tQWmOBW6FvhEw" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^b\\cdot\\:a^c$$", "result": "=\\sqrt{2^{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2}\\sqrt{2^{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTnGW/jTwUrqHAcqr1cHDsBC+r46Eolb4yfb3qibo0qjOCUCWbkwGOY7PqKo3U/JLJVmgCQUJbEmB6Cz9dIGB/Jujeh7+jKEzLb7VNCEMF3Z/sqxBQpByw1rOs2gJDdV//L292pD5hySoUKIRHggU0KNwuhBMtn2r3wONHUGucsPbJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+2\\sqrt{2}}{2\\cdot\\:1},\\:{u}_{2}=\\frac{-\\left(-2\\right)-2\\sqrt{2}}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+2\\sqrt{2}}{2\\cdot\\:1}:{\\quad}1+\\sqrt{2}$$", "input": "\\frac{-\\left(-2\\right)+2\\sqrt{2}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+2\\sqrt{2}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2+2\\sqrt{2}}{2}" }, { "type": "interim", "title": "Factor $$2+2\\sqrt{2}:{\\quad}2\\left(1+\\sqrt{2}\\right)$$", "input": "2+2\\sqrt{2}", "result": "=\\frac{2\\left(1+\\sqrt{2}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:1+2\\sqrt{2}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(1+\\sqrt{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=1+\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fv9gVNWyI/X8TvzzSZV8n9BT7cqoSAntC7uTSLRIMJq2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyB2rzZnucNvBU5dymMSHT2JBYjcLsCD0zopRg7yXiHFAz1ZIwC6YFMYnp61N6d4aXelnqK0S844f1w6EAcTrejRsZSEP36ANCDTv0XrTRe7l" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-2\\sqrt{2}}{2\\cdot\\:1}:{\\quad}1-\\sqrt{2}$$", "input": "\\frac{-\\left(-2\\right)-2\\sqrt{2}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-2\\sqrt{2}}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2-2\\sqrt{2}}{2}" }, { "type": "interim", "title": "Factor $$2-2\\sqrt{2}:{\\quad}2\\left(1-\\sqrt{2}\\right)$$", "input": "2-2\\sqrt{2}", "result": "=\\frac{2\\left(1-\\sqrt{2}\\right)}{2}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=2\\cdot\\:1-2\\sqrt{2}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(1-\\sqrt{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=1-\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yzZ+Rg5UPtRAM2q6UZuzBtBT7cqoSAntC7uTSLRIMJq2yYtEqeT7UjI2zAoMd7PXcJChiVhDxT5N/LHSTLMjyIcYUBmC3eatQPHmRkJsiutBYjcLsCD0zopRg7yXiHFAHY/HyZjN1C2qZGgDt8Y8hulnqK0S844f1w6EAcTrejRsZSEP36ANCDTv0XrTRe7l" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=1+\\sqrt{2},\\:u=1-\\sqrt{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=1+\\sqrt{2},\\:\\cos\\left(x\\right)=1-\\sqrt{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=1+\\sqrt{2}{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=1+\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=1-\\sqrt{2}{\\quad:\\quad}x=\\arccos\\left(1-\\sqrt{2}\\right)+2πn,\\:x=-\\arccos\\left(1-\\sqrt{2}\\right)+2πn$$", "input": "\\cos\\left(x\\right)=1-\\sqrt{2}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=1-\\sqrt{2}", "result": "x=\\arccos\\left(1-\\sqrt{2}\\right)+2πn,\\:x=-\\arccos\\left(1-\\sqrt{2}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=1-\\sqrt{2}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "x=\\arccos\\left(1-\\sqrt{2}\\right)+2πn,\\:x=-\\arccos\\left(1-\\sqrt{2}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arccos\\left(1-\\sqrt{2}\\right)+2πn,\\:x=-\\arccos\\left(1-\\sqrt{2}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.99787…+2πn,\\:x=-1.99787…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos(x)-\\cos^{2}(x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }